Transcript probability distribution. - McGraw Hill Higher Education

Discrete Probability
Distributions
Chapter 6
McGraw-Hill/Irwin
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
LO1 Identify the characteristics of a probability distribution.
LO2 Distinguish between discrete and continuous random
variable.
LO3 Compute the mean of a probability distribution.
LO4 Compute the variance and standard deviation of a
probability distribution.
LO5 Describe and compute probabilities for a binomial
distribution.
LO6 Describe and compute probabilities for a
hypergeometric distribution.
LO7 Describe and compute probabilities for a Poisson
distribution.
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LO1 Identify the characteristics
of a probability distribution.
What is a Probability Distribution?
PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated
with each outcome.
CHARACTERISTICS OF A PROBABILITY DISTRIBUTION
1.
The probability of a particular outcome is between 0 and 1 inclusive.
2.
The outcomes are mutually exclusive events.
3.
The list is exhaustive. So the sum of the probabilities of the various events is equal to 1.
Experiment:
Toss a coin three times. Observe the number of
heads. The possible results are: Zero heads, One
head, Two heads, and Three heads.
What is the probability distribution for the number
of heads?
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LO1
Random Variables
RANDOM VARIABLE A quantity resulting from an experiment that, by chance, can assume different values.
DISCRETE RANDOM VARIABLE A random
variable that can assume only certain clearly
separated values. It is usually the result of
counting something.
EXAMPLES
1.
The number of students in a class.
2.
The number of children in a family.
3.
The number of cars entering a carwash in a
hour.
4.
Number of home mortgages approved by
Coastal Federal Bank last week.
CONTINUOUS RANDOM VARIABLE can assume an
infinite number of values within a given range. It is
usually the result of some type of measurement
EXAMPLES
1.
The length of each song on the latest Tim McGraw
album.
2.
The weight of each student in this class.
3.
The temperature outside as you are reading this
book.
4.
The amount of money earned by each of the more
than 750 players currently on Major League Baseball
team rosters.
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LO3 and LO4 Compute the mean, standard
deviation and variance of a probability distribution.
The Mean and Variance of a Discrete
Probability Distribution
MEAN
•The mean is a typical value used to represent the central location of a probability distribution.
•The mean of a probability distribution is also referred to as its expected value.
VARIANCE AND STANDARD DEVIATION
• Measures the amount of spread in a distribution
• The computational steps are:
1. Subtract the mean from each value, and square this difference.
2. Multiply each squared difference by its probability.
3. Sum the resulting products to arrive at the variance.
The standard deviation is found by taking the positive square root of the variance.
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LO3, LO4
Mean, Variance, and Standard
Deviation of a Probability Distribution - Example
John Ragsdale sells new
cars for Pelican Ford. John
usually sells the largest
number of cars on
Saturday. He has
developed the following
probability distribution for
the number of cars he
expects to sell on a
particular Saturday.
MEAN
VARIANCE
STANDARD
DEVIATION
   2  1.290  1.136
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LO5 Describe and compute probabilities for
a binomial distribution.
Binomial Probability Distribution


1.
2.
3.
4.
A Widely occurring discrete probability distribution
Characteristics of a Binomial Probability Distribution
There are only two possible outcomes on a particular trial of an experiment.
The outcomes are mutually exclusive,
The random variable is the result of counts.
Each trial is independent of any other trial
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LO5
Binomial Probability Distribution
EXAMPLE
There are five flights daily from Pittsburgh via US Airways into the Bradford,
Pennsylvania, Regional Airport. Suppose the probability that any flight arrives
late is .20.
What is the probability that none of the flights are late today?
What is the average number of late flights? What is the variance of the
number of late flights?
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LO5
Binomial Distribution - Example
Binomial – Shapes for Varying  (n constant)
EXAMPLE
Five percent of the worm gears
produced by an automatic, high-speed
Carter-Bell milling machine are
defective.
What is the probability that out of six
gears selected at random none will be
defective? Exactly one? Exactly two?
Exactly three? Exactly four? Exactly
five? Exactly six out of six?
Binomial – Shapes for Varying n ( constant)
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Cumulative Binomial Probability Distributions Example
LO5
EXAMPLE
A study by the Illinois Department of Transportation concluded that 76.2
percent of front seat occupants used seat belts. A sample of 12 vehicles
is selected.
What is the probability the front seat occupants in exactly 7 of the 12
vehicles are wearing seat belts?
What is the probability the front seat occupants in at least 7 of the 12
vehicles are wearing seat belts?
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LO6 Describe and compute probabilities for
a hypergeometric distribution.
Hypergeometric Probability Distribution
1. An outcome on each trial of an experiment is classified into one of two mutually exclusive
categories—a success or a failure.
2. The probability of success and failure changes from trial to trial.
3. The trials are not independent, meaning that the outcome of one trial affects the outcome
of any other trial.
Note: Use hypergeometric distribution if experiment is binomial, but sampling is without
replacement from a finite population where n/N is more than 0.05
Formula:
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LO6
Hypergeometric Probability Distribution
EXAMPLE
PlayTime Toys, Inc., employs 50 people in the Assembly Department. Forty of the employees belong
to a union and ten do not. Five employees are selected at random to form a committee to meet with
management regarding shift starting times.
What is the probability that four of the five selected for the committee belong to a union?
Here’s what’s given:
N = 50 (number of employees)
S = 40 (number of union employees)
x = 4 (number of union employees selected)
n = 5 (number of employees selected)
6-12
LO7 Describe and compute
probabilities for a Poisson distribution.
Poisson Probability Distribution
The Poisson probability distribution describes the number of times some event occurs during a
specified interval. The interval may be time, distance, area, or volume.
Assumptions of the Poisson Distribution
(1)
The probability is proportional to the length of the interval.
(2)
The intervals are independent.
Examples include:
• The number of misspelled words per page in a newspaper.
• The number of calls per hour received by Dyson Vacuum Cleaner Company.
• The number of vehicles sold per day at Hyatt Buick GMC in Durham, North Carolina.
• The number of goals scored in a college soccer game.
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LO7
Poisson Probability Distribution - Example
EXAMPLE
Assume baggage is rarely lost by Northwest Airlines.
Suppose a random sample of 1,000 flights shows a
total of 300 bags were lost. Thus, the arithmetic
mean number of lost bags per flight is 0.3
(300/1,000). If the number of lost bags per flight
follows a Poisson distribution with u = 0.3, find the
probability of not losing any bags.
.
Use Appendix B.5 to find the probability that no
bags will be lost on a particular flight.
What is the probability exactly one bag will be lost
on a particular flight?
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LO7
More About the Poisson Probability Distribution
•The Poisson probability distribution is always positively skewed and the random variable has no
specific upper limit.
•The Poisson distribution for the lost bags illustration, where µ=0.3, is highly skewed.
•As µ becomes larger, the Poisson distribution becomes more symmetrical.
6-15